where \(C(t)\)  is the battery's coherence defined in Eq. (REF ), \(\tilde{\rho }_b(t)\)  is the passive state, \(\tau _\beta \)  is an arbitrary thermal state with the inverse temperature \(\beta \) , and \(D(\rho \Vert \sigma )=\mathrm {Tr}\lbrace \rho (\log \rho -\log \sigma )\rbrace \)  is the quantum relative entropy.
If the passive state can be rewritten as a thermal state, then coherent work is exactly the coherence up to a coefficient \(\beta ^{-1}\) .
This condition naturally holds for QBs with only two energy levels,
but it is not satisfied for general QBs.
Although it has been shown that \(W_e^c\)  is not a coherence monotone in general [1]}, we still find that the battery's coherence can qualitatively describe the change of coherent work.
Figures REF (b) and REF (d) show that coherent work \(W_e^c(t)\)  is positively related to coherence \(C(t)\) 
in the TC battery and the \(XXZ\)  battery.
